Barad's factorization conjecture for the type B Temperley–Lieb Gram determinant

Let ${\bf A}_n$ be an annulus with $2n$ marked points, and let ${\bf b}_n$ be the set of all diagrams, up to deformation, with $n$ non-crossing chords connecting these points. Define the bilinear pairing by gluing two diagrams along the marked circle; if the resulting annulus has $m$ homotopically non-trivial circles and $r$ homotopically trivial circles, set $\langle b_i,b_j\rangle=\alpha^m\delta^r$. Let

$$G_n(\alpha,\delta)=\Big(\langle b_i,b_j\rangle\Big)_{1\leqslant i,j\leqslant \binom{2n}{n}}$$

be the Gram matrix, and write its determinant as $D^B_n(\alpha,\delta)$. Let $T_i(\delta)$ be defined by

$$T_0=2,\qquad T_1=\delta,\qquad T_i=\delta T_{i-1}-T_{i-2}.$$

Barad's factorization conjecture. The determinant satisfies

$$D^B_n(\alpha,\delta)=\prod_{i=1}^n\left(T_i(\delta)^2-\alpha^2\right)^{\binom{2n}{n-i}}.$$

The factorization describes the complete set of predicted roots of the Gram determinant for the type B Temperley–Lieb algebra. The surrounding text attributes the prediction of the roots to Dąbkowski and Przytycki and the complete factorization to G. Barad; the parser supplies no evidence that the conjecture had been resolved in the source.